De Bruijn graphs and powers of $3/2$

Abstract

In this paper we consider the set ${\mathbb{Z}}_6^{\pm \omega }$ of two-way infinite words $\xi$ over the alphabet $\{0,1,2,3,4,5\}$ with the integer left part $\lfloor \xi \rfloor$ and the fractional right part $\{\xi \}$ separated by a radix point. For such words, the operation of multiplication by integers and division by $6$ are defined as the column multiplication and division in base 6 numerical system. The paper develops a finite automata approach for analysis of sequences ${\left(\lfloor \xi {\left(\frac{3}{2}\right)}^n\rfloor \right)}_{n\in \mathbb{Z}}$ for the words $\xi \in {\mathbb{Z}}_6^{\pm \omega }$ that have some common properties with $Z$-numbers in Mahler's $3/2$-problem. Such sequence of $Z$-words written under each other with the same digit positions in the same column is an infinite $2$-dimensional word over the alphabet ${\mathbb{Z}}_6$. The automata representation of the columns in the integer part of $2$-dimensional $Z$-words has the nice structural properties of the de Bruijn graphs. This way provides some sufficient conditions for the emptiness of the set of $Z$-numbers. Our approach has been initially inspirated by the proposition 2.5 in [1] where authors applies cellular automata for analysis of ${\left(\left\{\xi {\left(\frac{3}{2}\right)}^n\right\}\right)}_{n\in \mathbb{Z}}$, $\xi \in \mathbb{R}$.

23 pages, 6 figures (in russian)